i) Formulate the dual of the following problem:
ii) Check whether is a feasible solution to the primal and
is a feasible solution to the dual.
iii) Use duality to check whether is an optimal solution to the primal.
Let's go step by step to solve this problem numerically. --- ### Step (i): Formulate the Dual Problem The given primal problem is: \[ \text{Minimize } z = 9x_1 + 12x_2 + 15x_3 \] Subject to: \[ 2x_1 + 2x_2 + x_3 \geq 10 \] \[ 2x_1 + 3x_2 + x_3 \geq 12 \] \[ x_1 + x_2 + 5x_3 \geq 14 \] \[ x_1, x_2, x_3 \geq 0 \] #### Dual Formulation For a minimization problem with \(\geq\) constraints, the corresponding dual problem is a maximization problem with \(\leq\) constraints. - Let \( y_1, y_2, y_3 \) be the dual variables corresponding to the three constraints of the primal problem. The dual problem is: \[ \text{Maximize } w = 10y_1 + 12y_2 + 14y_3 \] Subject to: \[ 2y_1 + 2y_2 + y_3 \leq 9 \] \[ 2y_1 + 3y_2 + y_3 \leq 12 \] \[ y_1 ______ __________ ___ ________ ____ _______ _________ ___ ______ _________ _______.
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Which of the following statements are true and which are false? Give reasons for your answer.
a) In an LP model, the feasible solution space can be effected when redundant constraints are deleted.
b) If the primal LPP has an optimal solution, then the set of feasible solution to its dual is bounded.
c) In a simplex iteration, an artificial variable can be dropped all together from the simplex table once the variable becomes non basic.
d) The addition of a constant to all the elements of a payoff matrix in a two – person zero sum game can affect only the value of the game, not the optimal mix of the strategies.
e) There may be a balanced transportation problem without any feasible solution.
a) A marketing manager has 5 salespersons and 5 sales districts. Considering the capabilities of the salespersons and the nature of the districts, the marketing manager estimates the sales per month (in thousand 
) for each salesperson in each distinct as follows
Salespersons
| Districts | |||||
| 1 | 2 | 3 | 4 | 5 | |
| A | 32 | 38 | 40 | 28 | 40 |
| B | 40 | 44 | 28 | 21 | 36 |
| C | 41 | 27 | 33 | 30 | 37 |
| D | 22 | 38 | 41 | 36 | 36 |
| E | 29 | 33 | 40 | 35 | 39 |
Find an initial basic feasible solution for the following transportation problem using matrix-minima method. Also find the transportation cost.
Without sketching the region, check whether is in the convex hull of the points
and
. If it is in the region, write P as convex combination of A,B and C.
i) Formulate the dual of the following problem:
ii) Check whether is a feasible solution to the primal and
is a feasible solution to the dual.
iii) Use duality to check whether is an optimal solution to the primal.
a) A toy company manufactures two types of doll; a basic version-doll A and a deluxe version-doll B. Each doll of type B takes twice as long as to produce as one of type A, and the company would have time to make a maximum 2000 per day if it produce only the basic version. The supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy dress of which there are only 600 per day available. The company makes profit of ₹3 and 5 per doll respectively on doll A and B. How many of each should be produced per day in order to maximize profit? Solve this problem by graphical method.a) A toy company manufactures two types of doll; a basic version-doll A and a deluxe version-doll B. Each doll of type B takes twice as long as to produce as one of type A, and the company would have time to make a maximum 2000 per day if it produce only the basic version. The supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy dress of which there are only 600 per day available. The company makes profit of 3 and 5 per doll respectively on doll A and B. How many of each should be produced per day in order to maximize profit? Solve this problem by graphical method.
b) Find all the basic solutions of the following system:
Test the following set for convexity.